Introduction to quantum mechanics; with applications to chemistry

INTRODUCTION

TO QUANTUM MECHANICS

With Applications to Chemistry

BY


LINUS PAULING, PH.D., Sc.D.

Professor of Chemistry, California Institute of Technolouy

AND

E. BRIGHT WILSON, JR., PH.D.
AS~lOciate

Professor of Chemistry, Harvard

Uni~ersity

MeGRAW-HILL BOOK COMPANY,
NEW YORK AND LONDON

1935

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INCl.

COPYRIGHT,

1935,

BY THE


MCGRAW-HILL BOOK COMPANY, INC.

PRINTED IN THE UNITED STATES OF AMERICA

All rights reserved. This book, or
parts thereof, may not be reproduced
in any form without permission of
the publishers.
XVII

THE MAPLJiJ PRESS COMPANY, YORK, PA.

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PREFACE
In writing this book we have attempted to produce a textbook
of practical quantum mechanics for the chemist, the experi­
mental physicist, and the beginning student of theoretical
physics. The book is not intended to provide a critical discus­
sion of quantum mechanics, nor even to present a thorough

survey of the subject. We hope that it does give a lucid and
easily understandable introduction to a limited portion of
quantum-mechanical theory; namely, that portion usually
suggested by the name "wave mechanics," consisting of the
discussion of the Schrodinger wave equation and the problems
which can be treated by means of it. The effort has been made
to provide for the reader a means of equipping himself with a
practical grasp of this subject, so that he can apply quantum
mechanics to most of the chemical and physical problems which
may confront him.
The book is particularly designed for study by men without
extensive previous experience with advanced mathematics, such
as chemists interested in the subject because of its chemical
applications. We have assumed on the part of the reader, in
addition to elementary mathematics through the calculus, only
some knowledge of complex quantities, ordinary differential
equations, and the technique of partial differentiation. It
may be desirable that a book written for the reader not adept
at mathematics be richer in equations than one intended for
the mathematician; for the mathematician can follow a sketchy
derivation with ease, whereas if the less adept reader is to be
led safely through the usually straightforward but sometimes
rather complicated derivations of quantum mechanics a firm
guiding hand must be kept on him. Quantum mechanics is
essentially mathematical in character, and an understanding
of the subject without a thorough knowledge of the mathematical
methods involved and the results of their application cannot be
obtained. The student not thoroughly trained in the theory
of partial differential equations and orthogonal functions must
iii

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PREFACE

iv

learn something of these subjects as he studies quantum mechanIn order that he may do so, and that he may follow the
ics.
discussions given without danger of being deflected from the
course of the argument by inability to carry through some minor
step, we have avoided the temptation to condense the various

and perhaps more elegant forms.
After introductory chapters on classical mechanics and the
old quantum theory, we have introduced the Schroding^r wave
discussions into shorter

equation and its physical interpretation on a postulatory basis,
and have then given in great detail the solution of the wave
equation for important systems (harmonic oscillator, hydrogen
atom) and the discussion of the wave functions and their properomitting none of the mathematical steps except the most
A, similarly detailed treatment
has been given
in the discussion di pert in option Shruor^, the variation method,

ties,

l^JT!^?577

1

the structure of simple molecules, and, in general,

'iu

--,

important section of the book.
In order to limit the size of the book, we have omitted from
discussion such advanced topics as transformation theory and
general

quantum mechanics

(aside

from

brief

mention

in the

chapter), the Dirac theory of the electron, quantization
of the electromagnetic field, etc.
We have also omitted several
last

subjects which are ordinarily considered as part of elementary
quantum mechanics, but which are of minor importance to the

chemist, such as the Zeeman effect and magnetic interactions in
general, the dispersion of light and allied phenomena, and
most of the theory of aperiodic processes.

The authors are severally indebted to Professor A. Sommerfeld
and Professors E. U. Condon and H. P. Robertson for their

own

quantum mechanics. The constant advice
Tolman is gratefully acknowledged, as well
Professor P. M. Morse, Dr. L. E. Sutton, Dr.

introduction to

of Professor R. C.

as the aid of

G.

W. Wheland,

Dr. L. 0. Brockway, Dr.

J.

Sherman, Dr.

S.

Weinbaum, Mrs. Emily Buckingham Wilson, and Mrs. Ava
Helen Pauling.

LINUS PAULING.
E. BRIGHT WILSON, JR.
PASADENA, I^AMF.,

CAMBRIDGE MASS.,
July, 193J5.

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CONTENTS
PREFACE

CHAPTER

I

SURVEY OF CLASSICAL MECHANICS
SECTION
1.

Newton's Equations

la.

of

Motion

The Three-dimensional

in the

Lagntngian Form

Isotropic

Harmonic

.....

Oscillator.

.......... \.

A

.

Ib.

Generalized Coordinates

Ic.

The Invariance of the Equations of Motion in the Lagraibgian Form
An Example: The Isotropic Harmonic Oscillator in Polar

.

.

.

.

....................

Id.

7

....................
of Angular Momentum
The Equations of Motion in the Hamiltonian Form ......
2a. Generalized Momenta .................

14

.......

16

Coordinates

le.

2.

2

4
6

26.

The Conservation

.

The Hamiltonian Function and Equations
The Hamiltonian Function and the Energy
A General Example

.

.

9
11

14

3.

......
...............
......
The Emission and Absorption
Radiation.

21

4.

Summary

23

2c.

2d.

of

of

Chapter

I

.

............
CHAPTER

.......

16

17

II

THE OLD QUANTUM THEORY
5.

The

5a.
56.
5c.
6.

7.

...........

Old Quantum Theory
The Postulates of Bohr
The Wilson-Sommerfeld Rules of Quantization ......
Selection Rules.
The Correspondence Principle .....

Origin of the

................

..............

The Quantization of Simple Systems
6a. The Harmonic Oscillator.
Degenerate States ......
66. The Rigid Rotator
6c. The Oscillating and Rotating Diatomic Molecule .....
6d. The Particle in a Box
6e. Diffraction by a Crystal Lattice
The Hydrogen Atom

..................
.................
...........
....................
7a. Solution of the Equations of Motion ..........
Application of the Quantum Rules. The Energy Levels
Description of the Orbits ...............
7d. Spatial Quantization .................
The Decline of the Old Quantum Theory ............
76.

.

7c.

8.

v

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.

25
26
28
29
30
30
31

32
33
34
36
36
39
43
45

47


CONTENTS

vi

PAGE

SECTION

CHAPTER

III

THE SCHRODINGER WAVE EQUATION WITH THE
HARMONIC OSCILLATOR AS AN EXAMPLE
9.

The Schrodinger Wave Equation
9a. The Wave Equation Including
96. The Amplitude Equation

50
the

Time

53
56

9c.

Wave

9d.

Energy Values
The Complex Conjugate Wave Function ty*(x,t)....

Discrete and Continuous Sets of Charac-

Functions.

58

teristic

10.

Physical Interpretation of the Wave Functions
t) as a Probability Distribution Function.
106. Stationary States

The

10a. Sk* (x, t)V(x,

lOc.

11.

63
63

63
64

Further

Physical

Interpretation.

Average

Values

Dynamical Quantities
The Harmonic Oscillator in Wave Mechanics
11 a. Solution of the

Wave Equation

116.

The Wave Functions

lie.

Mathematical Properties

for the

Harmonic

Oscillator

...

Physical Interpretation

of the

.

and

of

.

.

.

.

their

73

.

Harmonic

65
67
67

Oscillator

Wave

Functions

77

CHAPTER IV

THE WAVE EQUATION FOR A SYSTEM OF POINT
PARTICLES IN THREE DIMENSIONS

15.

The Wave Equation for a System of Point Particles
12a. The Wave Equation Including the Time
126. The Amplitude Equation
12c. The Complex Conjugate Wave Function ty*(xi
ZAT, t)
12d The Physical Interpretation of the Wave Functions ...
The Free Particle
The Particle in a Box
The Three-dimensional Harmonic Oscillator in Cartesian Coordi-

16.

Curvilinear Coordinates

17.

The Three-dimensional Harmonic

12.

13
14.

84
85
86
88
88
90
95
100

nates

103
Oscillator in Cylindrical Coordi-

nates

105

CHAPTER V

THE HYDROGEN ATOM
18.

The

Solution of the

Wave Equation by

and the Determination
18a.

The Separation

186.

The

tional

of

of the

the

the Polynomial

Wave

Equation.

Motion

Solution of the

*>

Method
113

Energy Levels

Equation

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The Transla113
117


CONTENTS

Vit

PAGE
118

SUCTION
18c.

18d.

The Solution of the & Equation
The Solution of the r Equation
The Energy Levels

121

124
125
Legendre Functions and Surface Harmonics
19a. The Legendre Functions or Legendre Polynomials
126
196. The Associated Legendre Functions
127
20. The Laguerre Polynomials and Associated Laguerre Functions
.129
20a. The Laguerre Polynomials
129
206. The Associated Laguerre Polynomials and Functions ... 131
21. The Wave Functions for the Hydrogen Atom
132
132
21a. Hydrogen-like Wave Functions
216. The Normal State of the Hydrogen Atom
139

142
21c. Discussion of the Hydrogen-like Radial Wave Functions.
2ld. Discussion of the Dependence of the Wave Functions
18c.

19.

.

.

on the Angles

t?

and

146


CHAPTER VI

PERTURBATION THEORY
Expansions in Series of Orthogonal Functions
23\xFirst-order Perturbation Theory for a Non-degenerate Level ...
23a. A Simple Example: T>/> P^^tujj^ejjJH^mjinnin QsillRf.nr
?jfo
An Exjunplp- Tfor Normal FHium Atn
24. J$tst-order Perturbation Theory for a Degenerate Level

24a. An Example: Application of a Perturbation to a Hydrogen
22.

.

Atom
25. Second-order Perturbation

25a.

156
160
162
165
172
176

Theory

An Example: The Stark

151

Effect of the Plane Rotator

.

.177

.

CHAPTER VII

THE VARIATION METHOD AND OTHER APPROXIMATE
METHODS
26.

The Variation Method
26a. The Variational Integral and its Pmpgrtjes
266. An Example fThe Normal State of
Helium Atom
the_
26c. Application of the Variation Method to Other States

.

.

.

...

26d. Linear Variation Functions

27.

26e. A More General Variation Method
Other Approximate Methods

27a.

A

276.

The Wentzel-Kramers-Brillouin Method

Generalized Perturbation Theory

27c. Numerical Integration
27d. Approximation by the Use of Difference Equations
270.

An Approximate

....

Second-order Perturbation Treatment

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.

.

180
180
184

186

186
189
191
191
198
201
202

204


CONTENTS

via

PAQB

SUCTION

CHAPTER VIII

THE SPINNING ELECTRON AND THE PAULI EXCLUSION
THE HELIUM ATOM

PRINCIPLE, WITH A DISCUSSION OF
28.
29.

The Spinning Electron
207

210
The Helium Atom. The Pauli Exclusion Principle
210
29a. The Configurations Is2s and Is2p
The Pauli Exclusion
296. The Consideration of Electron Spin.
214

Principle

The Accurate Treatment of the Normal Helium Atom.
Excited States of the Helium Atom
The Polarizability of the Normal Helium Atom

29c.

29d.
29e.

.

.

221
225
226

CHAPTER IX

MANY-ELECTRON ATOMS

30. Slater's

30a.
306.
30c.

Treatment of Complex Atoms
Exchange Degeneracy
Spatial Degeneracy
Factorization and Solution of the Secular Equation.

...

31 a.

32.

Applications.

The Lithium Atom and Three-electron Ions

316. Variation

The Method

Treatments

of

Other Atoms.

.

.

.

.

.

....

.

.

of the Self-consistent Field

32a. Principle of the Method
326. Relation of the Self-consistent Field
tion Principle
32c. Results of the Self-consistent Field
33.

.

*

30d. Evaluation of Integrals

30e. Empirical Evaluation of Integrals.
31. Variation Treatments for Simple Atoms

Method

to the Varia-

252
254
256
256
257

Method

Other Methods for Many-electron Atoms
33a. Semi-empirical Sets of Screening Constants
336.

The Thomas-Fermi

Statistical

CHAPTER

230
230
233
235
239

244
246
247
249
250
250

Atom

X

THE ROTATION AND VIBRATION OF MOLECULES
34.
35.

The Separation of Electronic and Nuclear Motion
259
The Rotation and Vibration of Diatomic Molecules
263
35a. The Separation of Variables and Solution of the Angular
Equations
356.

The Nature

35.

A Simple Potential Function for Diatomic
A More Accurate Treatment. The Morse

35d.
36.

of the Electronic

Energy Function
Molecules
Function

The Rotation of Polyatomic Molecules
36a. The Rotation of Symmetrical-top Molecules
36b. The Rotation of Unsymmetrical-top Molecules

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.

.

.

.

.

.

264
266
267

271

275
275
280


CONTENTS

ix

SECTION
37.

38.

PAGE
282
282
288
290

The Vibiation of Polyatomic Molecules
37a. Normal Coordinates in Classical Mechanics
376. Normal Coordinates in Quantum Mechanics
The Rotation of Molecules in Crystals

CHAPTER XI

PERTURBATION THEORY INVOLVING THE TIME, THE

EMISSION AND ABSORPTION OF RADIATION, AND THE

RESONANCE PHENOMENON
39.

The Treatment

of a

Time-dependent Perturbation by the Methoa

....
....
....
....

of Variation of Constants

294
296
299
299

39a.

A

40a.

The Einstein Transition Probabilities

The Calculation of the Einstein Transition Probabilities
302
by Perturbation Theory
Selection Rules and Intensities for the Harmonic Oscillator 306

Simple Example
40. The Emission and Absorption of Radiation .V
406.

40c.

and Intensities for Surface-harmonic Wave
Functions
306
40e. Selection Rules and Intensities for the Diatomic MoleculevThe Franck-Condon Principle
309
.312
40/. Selection Rules and Intensities for the Hydrogen Atom

40d. Selection Rules

.

Even and Odd Electronic
The Resonance Phenomenon
400.

41.

States and their Selection Rules. 313

314
315
318
322

Resonance in Classical Mechanics
416. Resonance in Quantum Mechanics
41c. A Further Discussion of Resonance

41a.

CHAPTER XII

THE STRUCTURE OF SIMPLE MOLECULES
42.

The Hydrogen Molecule-ion

327
327
Very Simple Discussion
426. Other Simple Variation Treatments
331
42c. The Separation and Solution of the Wave Equation .... 333
42d. Excited States of the Hydrogen Molecule-ion
340
42a.

43.

A

The Hydrogen Molecule
43a. The Treatment of
43c.

and London.
Other Simple Variation Treatments
The Treatment of James and Coolidge

43d.

Comparison with Experiment

436.

.

.

43e. Excited States of the
43/. Oscillation

340
340
345
349

Heitler

.

351

Hydrogen Molecule

and Rotation

of

353

Molecule.

Ortho and

Interaction of

Two Normal

the

Para Hydrogen
44.

The Helium Molecule-ion Hef and the
Helium Atoms
44a. The Helium Molecule-ion Hef

355

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358
358


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INTRODUCTION TO QUANTUM
MECHANICS
CHAPTER

I

SURVEY OF CLASSICAL MECHANICS
The subject of quantum mechanics constitutes the most recent
step in the very old search for the general laws_goyrning the
motion of matter. For a long time investigators confined their
studying the dynamics of bodies of macroscopic dimenwhile the science of mechanics remained in that
and
sions,
was
it
stage
properly considered a branch of physics. Since
the development of atomic theory there has been a change of
emphasis. It was recognized that the older laws are not correct

when applied to atoms and electrons, without considerable
modification.
Moreover, the success which has been obtained
efforts to

making the necessary modifications of the older laws has also
had the result of depriving physics of sole claim upon them, since
it is now realized that the combining power of atoms and, in
fact, all the chemical properties of atoms and molecules are
in

explicable in terms of the laws governing the motions of the
electrons and nuclei composing them.

Although it is the modern theory of quantum mechanics in
which we are primarily interested because of its applicatiqns_to
chemical jjroblems it is desirable for us first to discuss briefly
the background of classical mechanics from which it was developed.
By so doing we not only follow to a certain extent the
historical development, but we also introduce in a more familiar
,

form

many

concepts which are retained in the later theory. We
problems in the first few chapters by the

shall also treat certain

methods

of the older theories in preparation for their later treatIt is for this reason that the

ment by quantum mechanics.

advised to consider the exercises of the first few
chapters carefully and to retain for later reference the results

student

is

which are secured.
1

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SURVEY OF CLASSICAL MECHANICS

2

[1-1

chapter no attempt will be made to give any parts
of classical dynamics but those which are useful in the treatment
With this restriction, we
of atomic and molecular problems.

of the dynamics of rigid
discussion
have felt justified in omitting

In the

first

bodies, non-conservative systems, non-holonomic systems, systems involving impact, etc. Moreover, no use is made of

Hamilton's principle or of the Hamilton-Jacobi partial differential
By thus limiting the subjects to be discussed, it is
equation.
possible to give in a short chapter a thorough treatment of

Newtonian systems
1.

of point particles.

NEWTON'S EQUATIONS OF MOTION IN THE LAGRANGIAN

FORM
The
is

earliest

formulation of dynamical laws of wide application
If we adopt the notation #-, y

Z{

that of Sir Isaac Newton.

t,

three Cartesian coordinates of the iih particle with
mass Wi, Newton's equations for n point particles are
for the

mx =
t

i

where

Z

1, 2,

-

-

-

,

n,

(1-1)

components of the force acting on
a set of such equations for each
Dots refer to differentiation with respect to time, so

X,-, F;,

t

are the three

the ^th particle.
particle.

=

There

is

that

*

-

v

<'- 2>

By introducing certain familiar definitions we change Equation
1-1 into a form which will be more useful later.
We define as
the kinetic energy

T

(for Cartesian coordinates) the

quantity

T =

If

we

tive

limit ourselves to a certain class of systems, called conservait is possible to define another quantity, the potential
V, which is a function of the coordinates x \y\z\
of all the particles, such that the force components acting

systems,

energy

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NEWTON'S EQUATIONS OF MOTION

1-1]

3

on each

particle are equal to partial derivatives of the potential
with
respect to the coordinates of the particle (with
energy

negative sign)

;

that

is,

Y At

>

7:

OXi

dV

Yi

=

7
Zi

dV
- ___,
_

=

i

~^Y

1, 2,

,

n.

(1-4)

It is possible to find a function V which will express in this manner
forces of the types usually designated as mechanical, electrostatic,

and gravitational. Since other types of forces (such as electromagnetic) for which such a potential-energy function cannot
be set up are not important in chemical applications, we shall
not consider them in

With these

detail.

Newton's equations become

definitions,

at dxi

dxi

dT

+

dt dyi

d

3V
dy,

dV =

dT

T
~^- + 3
dt dZi
dZi
t

.
n
'
(l-5c)
^
.

n
0.

There are three such equations for every particle, as before.
These results are definitely restricted to Cartesian coordinates;
but by introducing a new function, the Lagrangian function L,
defined for Newtonian systems as the difference of the kinetic

and potential energy,

L =

L(XI,

i/i, 21,

,

xn yn
,

we can throw the equations
prove to be valid in
Cartesian coordinates

later

In

,

of

z n , Xi f

-

,

=
T -

F,

(1-6)

motion into a form which we shall

any system
7

T

z n)

is

a

of coordinates (Sec. Ic).

function of the velocities

and for the systems to which our treatment
a function of the coordinates only; hence the
equations of motion given in Equation 1-5 on introduction of
the function L assume the form
xi,
is

,

z n only,

restricted

F

is

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SURVEY OF CLASSICAL MECHANICS
d^dL

P-la

_ dL =
'

eft

to*

dZi

=

t

_

d

_

d

=

1, 2,

,

n.

(1-7)

'

In the following paragraphs a simple dynamical system
by the use of these equations.

is

discussed
la.

As an

The Three-dimensional

Harmonic Oscillator.
motion in this

Isotropic

illustration of the use of the equations of

we choose a system which has played a very important
part in the development of quantum theory. This is the
lwrn,wiwscj$M&r, a particle bound to an equilibrium position by
form,

a force which increases in magnitude linearly with its distance
r from the point.
In the three-dimensional isotropic harmonic
oscillator this corresponds to a potential function %kr 2 represent,

ing a force of magnitude kr acting in a negative direction; i.e.,
from the position of the particle to the origin, k is called the
force constant or

nates

Hookas-law

Using Cartesian coordi-

constant.

we have

L =

(1-8)

whence
-r.(mx)

+

fcx

flf

= mx

+ fcx ==
my + ky =
=
wz +
fcz

Multiplication of the

first

member

of

0,
0,

(1-9)

0.

Equation 1-9 by x gives

~ ~ kx dx
mx.dx _
,

dt

~dt

(1-10)

or

2

dt

dl

(1-11)

which integrates directly to

^mx =

2

The constant

of integration

%kx 2
is

+

constant.

conveniently expressed as

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(1-12)


NEWTON'S EQUATIONS OF MOTION

I-la]

Hence

^mvl

on introducing the expression
constant k,

or,

in place of the force

dx

which on integration becomes

+

27n>o*

=

dx

sin" 1

XQ

or

x

=

x sin (27r^

+

(1-14)

$*),

and similarly
t/

Z

In these expressions

=
=
rr

t/o

sin (Zirwt

Z

Sin

,

(27T*>o

),)
z ).

(1-15)

/

6 y and d z are constants of
Zo, 5 X
which determine the motion in any

2/0,

integration, the values of

,

The quantity VQ is
by the equation

given case.

+
+S
,

related to the constant of the

restoring force

Wmvl =

(1-16)

k,

so that the potential energy

may be written
V = 27r 2m^r 2

as

(1-17)

.

As shown by the equations
the motion.

for x,

It is seen that

T/,

and

z,

the particle

j>

is

may

the frequency of
be described as

carrying out independent harmonic oscillations along the x,

and

z axes,

with different amplitudes

XQ,

2/o,

and

ZQ

and

y,

different

phase angles 6 X d y and d z respectively.
The energy of the system is the sum of the kinetic energy and
the potential energy, and is thus equal to
,

On evaluation,
value

2

27r mj/

it is

2
)

,

2
(:r

)

+

,

found to be independent of the time, with the

determined by the amplitudes of
2/o + z l)

oscillation.

The one-dimensional harmonic oscillator, restricted to -motion
along the x axis in accordance with the potential function
F = %kx 2 = 27r 2rai>Jz 2 , is seen to carry out harmonic oscillations

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SURVEY OF CLASSICAL MECHANICS

6

along this axis as described by Equation 1-14.
2
is given by the expression 2T mj>gzJ.
Ib. Generalized Coordinates.

nates
to use

xij

2/1, 2i,

xn

,

some other -set

,

7/ n ,

[I-lb

Its total

energy

Instead of Cartesian coordi-

n, it is

frequently more convenient

of coordinates to specify the configuration

For example, the isotropic spatial harmonic
oscillator already discussed might equally well be described using
polar coordinates; again, the treatment of a system composed of
two attracting particles in space, which will be considered
later, would be very cumbersome if it were necessary to use

of the system.

rectangular coordinates.
If we choose, any set of 3n coordinates, which we
assume to be independent and at the same time

shall

always

sufficient in

number

to specify completely the positions of the particles of
the system, then there will in general exist 3n equations, called
the equations of transformation, relating the new coordinates
Qk to the set of Cartesian coordinates x tj y lf z t;

(1-18)

There

is

such a set of three equations for each particle

i.

The

functions / t g iy hi may be functions of any or all of the 3n new

coordinates q^ so that these new variables do not necessarily
For example,
split into sets which belong to particular particles.
,

two particles the six new coordinates may be the
Cartesian coordinates of the center of mass together
with the polar coordinates of one particle referred to the other

in the case of

three

particle as origin.

As

is

known from

the theory of partial differentiation,

possible to transform derivatives

from one

it is

set of

independent
variables to another, an example of this process being

=
dt

dq, dt

^

.

dq 2 dt

.

.

"*"

This same equation can be put in the

+

dq* n ~3T'

I

much more compact form

3n

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NEWTON'S EQUATIONS OF MOTION

I-lc]

7

This gives the relation between any Cartesian component of
Similar
velocity and the time derivatives of the new coordinates.
The
relations, of course, hold for t/ and 2< for any particle.
quantities g/, by analogy with x if are called generalized velocities,
even though they do not necessarily have the dimensions of
length divided by time (for example, Since partial derivatives transform in just the same manner,
we have

_
dxi dqj
d

_

.

_
dz n d

dyi dqj

Z **< + *Z to* + dV dz <\
dz>
.

.

dx, dqj

dyi dq]

dqj

given by an expression in terms of V and that for the force Xi in terms of V and #,, it is called
to
analogous
Since Q,

is

a generalized force.
In exactly similar fashion, we have
L

.

T dqj

Ic,

_J_

dyi dqj

The Invariance of the Equations
We are now in a position

gian Form.

of

to

Motion in the Lagranshow that when New-

ton's equations are written in the form given by Equation 1-7
they are valid for any choice of coordinate system. For this

we

apply a transformation of coordinates to Equamethods of the previous section. Multiplicadx
dlJ
*> of 1-56 by
tion of Equation l-5a by

^> etc., gives

proof

shall

tions 1-5, using the

'

'

oOj

uQj

ox\ a d J.

-^^v^at dxi

+

dv

dx\

^

dqj

^
ox\ dqj

dqj dt dx%

dx% dqj

^

'

I

dqj dt

x

=o,

5 ==

n

dx n

with similar equations in y and

(122)

dx n

z.

dq,-

Adding

gives

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all

of these together


SURVEY OF CLASSICAL MECHANICS

dW fadW ^ faddQ ^ dV
"*"

qj dt dXi

dq, dt diif

dq, dt dyi

[I-lc

=

(
V

'

'

dq f

Equation 1-20 has been used. In order to
sum, we note the following identity, obtained by

where the

result of

reduce the

first

differentiating a product,

=

_i

dqjdt\dXi)

i

_

(

obtain directly

Furthermore, because the order of differentiation

we

is

immaterial,

see that

d/dXi\

By

}

dXidt\d'

dt\dXidqjJ

From Equation l-19b we

i

_

i

d /dXi\

introducing Equations 1-26 and 1-25 in 1-24 and using the

result in

Equation 1-23, we get

d/dTdi .dTdyi
^
dt\dXi dqj

dyi

3%

,

"^

^T <^\

/dTdXi

dz*

\dxi dqj

dqj

dTdy*
"*"

dyi dq^

which, in view of the results of the last section, reduces to

IJS

-+-

<>*

=
y
Finally, the introduction of the Lagrangian function
V a function of the coordinates only, gives the more compact

L

T

V

with

form

--

'

^

= 1^,3,-..,

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3n.

(1-29)


NEWTON'S EQUATIONS OF MOTION

I-ld]

(It is

important to note that

of the coordinates

Since the above

L must

9

be expressed as a function

and

their first time-derivatives.)
derivation could be carried out for

any value

there are 3n such equations, one for each coordinate g/.
,/,
They are called the equations of motion in the Lagrangian form
of

and are of great importance. The method by which they were
derived shows that they are independent of the coordinate
system.

We have so far rather limited the types of systems considered,
but Lagrange's equations are much more general than we have
indicated and by a proper choice of the function L nearly all dynamiThese equations are
cal problems can be treated with their use.
therefore frequently chosen as the fundamental postulates of
classical mechanics instead of Newton's laws.
Id. An Example The Isotropic Harmonic Oscillator in Polar
Coordinates. The example which we have treated in Section la
can equally well be solved by the use of polar coordinates r,

The equations of transformation correspond#, and $ (Fig. 1-1)
ing to Equation 1-18 are
:

.

=
=
=

x
y
z

rsin#cosr sin

sin

tf

$>,

rcostf.

\

(1-30)

j

With the use of these we find for the kinetic and
of the isotropic

T =

~

V =

27r

m(x*
2

harmonic

+ y* + * = | (r + rW + r

mv 2 r 2

potential energies

oscillator the following expressions:

2

2

)

2

sin

,

and

L = T - V = ~(r 2
Z
The equations

of

+ rW + r

d$

do-

at

or

at

at or

sin 2 tty 2 )

-

27r

2

m^r 2

(1-32)

.

motion are
2

at

2

^-

^

'

mr* sin * cos

- mr sin 2

W=
W+

(1-34)

0,

^r^mv\r

-

0.

(1-35)

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SURVEY OF CLASSICAL MECHANICS

10

I-td

In Appendix II it is shown that the motion takes place in a
plane containing the origin. This conclusion enables us to
Let us
simplify the problem by making a change of variables.
that
at

such
the
time
coordinates
introduce new polar
r, #', x

=

the plane determined by the vectors r and v, the position
and velocity vectors of the particle at t = 0, is normal to the new
This transformation is known in terms of the old set of
z' axis.

t

coordinates
of the axis

FIG. 1-1.

dinates

if

z'

The

r, t?

,

two parameters # and

in

v?

,

determining the position

terms of the old coordinates, are given

relation of polar coor

The

Fio. 1-2.

(Fig. 1-2).

rotation of axes.

and

coordinates, the Lagrangian function L
motion have the same form as previously,
because the first choice of axis direction was quite arbitrary.

However, since the coordinates have been chosen so that the
plane of the motion is the x'y' plane, the angle #' is always equal

In terms of the

new

and the equations

of

&

to a constant, v/2.
in Equation 1-33
Inserting this value of
and writing it in terms of x instead of
-

0,

(1-36)

a constant.

(1-37)

:)"

U*/

which has the solution

mr*x

The

r equation,

=

Px>

Equation 1-35, becomes
oL
dt

=

(

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0,


NEWTON'S EQUATIONS OF MOTION

Me]

or,

11

using Equation 1-37,

mr*a

an equation

differing

-

.

.

.

=

w

v
(1-38)
y

0,'

from the related one-dimensional Cartesian3
p|/wr which

coordinate equation by the additional term
represents the centrifugal force.

Multiplication by r and integration with respect to the time
gives

so that r

/

=

02
~~

m

I

\

2

r

T -

=

r2

^*

2

-

2

2

47r v u r

ra 2 r 2

2

+ 6,
i

(1-39)
\
/

>

V*

47r

2

^r

+

2

6

I

/

This can be again integrated, to give
rdr

+
which x

=

2

bx

2

a
p%/m b
2
=
47r
and
c
Equation 1-39,
which yields the equation

in

r

,

,

is
2

j> ,.

-^-J6
with

A

+A

+

cx*)*'

the constant of integration in
This is a standard integral

sin

given by

We

have thus obtained the dependence of r on the time, and
by integrating Equation 1-37 we could obtain x as a function of
the time, completing the solution. Elimination of the time
between these two results would give the equation of the orbit,
which is an ellipse with center at the origin. It is seen that the

constant
le.

again occurs as the frequency of the motion.

v

The Conservation

of

Angular Momentum.

The example

previous section illustrates an important
of
wide
the principle of the conservation
applicability,
principle

worked out
of angular

in the

momentum.

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SURVEY OF CLASSICAL MECHANICS

12

[I-le

Equation 1-37 shows that when x is the angular velocity of the
particle about a fixed axis z' and r is the distance of the particle
from the axis, the quantity px = wr 2 x is a constant of the motion. 1

This quantity is called the angular momentum of the particle
about the axis z'.
It is not necessary to choose an axis normal to the plane of the
motion, as

Thus

in this example, in order to

z'

integrable to
rar 2 sin 2 &
Here

apply the theorem.
z, is at once

Equation 1-33, written for arbitrary direction

r sin

&

= pvy

a constant.

(1-40)

the distance of the particle from the axis z, so that
equation is the angular momentum about the

is

left side of this

the
axis

z.

2

It is

seen to be equal to a constant, p^.

FIG. 1-3.

Figure showing the relation between dx, d&, and

d
In order to apply the principle, it is essential that the axis of
Thus the angle d of polar coordinates
has associated with it an angular momentum p# = mr 2 $ about

reference be a fixed axis.

an axis in the xy plane, but the principle of conservation of
angular momentum cannot be applied directly to this quantity
because the axis is not, in general, fixed but varies with simple relation involving p* connects the angular
1

The phrase a constant

of the motion is often used in referring to a constant
equations of motion for a dynamical system.
sometimes referred to as the component of angular momentum

of integration of the
2

This

is

momenta

along the axis

z.

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NEWTON'S EQUATIONS OF MOTION

I-le]

px and

13

about different fixed axes, one of which, px relates
This is

p
,

to the axis normal to the plane of the motion.

= Pd$

Pxdx

+ Pvdv,

(1-41)

The
easily derived by considering Figure 1-3.
the
small
sides of
triangle have the lengths r sin M>, rdx, and

Since they form a right triangle, these distances are
rd&.
connected by the relation
an equation

=

rL

sm*

which gives, on introduction
and multiplication by m/dt>

mr^xdx

=

wi?*

of the angular velocities x,

2

sin 2 $
+

r/ir

2

<f>,

and #

$d$.

Equation 1-41 follows from this and the definitions of p x p*,
and pv
Conservation of angular momentum may be applied to more
It is at once
general systems than the one described here.
evident that we have not used the special form of the potential,

.

energy expression except for the fact that

it is

independent of

direction, since this function enters into the r equation only.
Therefore the above results are true for a particle moving in

any

spherically symmetric potential

field.

Furthermore, we can extend the theorem

to a collection of
with
each
other
in
point particles interacting
any desired way
but influenced by external forces only through a spherically
symmetric potential function. If we describe such a system by
using the polar coordinates of each particle, the Lagrangian

function

is

n

L =
Instead of


coordinates a,

^2 5/Wi(^?

^2

,

,

0,

*

,



=
-

oi

+ ?*&* + r

2

sin 2

we now

introduce

new angular

given by the linear equations

+ 6i0 +
+
+
2 /3

+ kiK,
+k

2 K,

_

(i 43)
J

The

values given the constants 61,
k n are unimportant so
long as they make the above set of equations mutually independ,

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SURVEY OF CLASSICAL MECHANICS

14

such that if a is increased
the
effect is to increase each
constant,
by Aa, holding /?,
to
rotate
the
whole
in
other
words,
system of particles

ent.

a

[I-2a

is

an angle about the axis
,

about

z

z

K

without changing their mutual positions. By hypothesis
V is not changed by such a rotation, so that V is

the value of

independent of

a.

We

therefore obtain the equation

ddL^dL^ddT^
da

dtda

U

dtda

**'

Moreover, from Equation 1-42 we derive the relation

t

-

1=1

1

Hence, calling the distance r sin
we obtain the equation
l

t>

t

of the ith particle

from the

z axis pi,

itpfci

=

constant.

(1-46)

This is the more general expression of the principle of the conservation of angular momentum which we were seeking.
In
such a system of many particles with mutual interactions, as,

an atom consisting of a number of electrons and a
the
individual particles do not in general conserve
nucleus,
momentum
but the aggregate does.
angular
for example,

The potential-energy function V need be only cylindrically
symmetric about the axis z for the above proof to apply,
since the essential feature was the independence of V on the angle
a about z. However, in that case z is restricted to a particular
direction in space, whereas if V is spherically symmetric the
theorem holds for any choice of axis.
Angular momenta transform like vectors, the directions of the
vectors being the directions of the axes about which the angular

momenta

are determined.

It is

customary to take the sense

of the vectors such as to correspond to the right-hand screw rule.

3.

THE EQUATIONS OF MOTION IN THE HAMILTONIAN FORM
2a. Generalized

momentum

Momenta.

In

Cartesian

related to the direction x k

is

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mx
k

k,

coordinates

which, since

the

V

is